An array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. It then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterised by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence.
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Second Edition S.M. Lane Categories for the Working Mathematician "A very useful introduction to category theory."—INTERNATIONALE MATHEMATISCHE NACHRICHTEN
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44 of 47 people found the following review helpful
Format:Hardcover
This book is a classic. Clearly written, drawing on a vast number of different applications and motivations for the subject. Eilenberg and Mac Lane created category theory and this book is alive with the very style of thought Mac Lane brought to it in the first place. It is obvious that Mac Lane wrote each page, and each exercise, with a view of the whole book in mind. He starts with the very basics, assuming indeed that you know nothing of category theory. He goes on to adjunctions, limits, the adjoint functor theorems, monads (triples), monoidal categories, Abelian cateories, Kan extensions, higher dimensional categories, and categorical foundations. It is a masterpiece and one of the great books in mathematics.
39 of 42 people found the following review helpful
Format:Hardcover|Amazon Verified Purchase
Well, let us think about this a little bit...You want to learn Category theory, whether for some course or just for the fun of it, and now where do you turn in order to learn the necessary concepts. If you are a mathematician and have some experience, then you turn to the masters, the originators of the given subject and read their work. Sure, being the founder of a given subject does not imply that you are a good expositor and hence are capable of revealing the necessary concepts for the beginner-allow me to inform that Mac Lane is indeed as good as an expositor as he was a mathematician. For any doubters, I point you to the only other text you should read on Category theory, namely, "Category Theory" by Horst Herrlich and compare this text with Mac Lane's. Aside from that, and with respect to the text, for most beginners or interested readers I would suggest the following outline: Read 1.1-6; 2.1-3 & 8 possibly 2.4; all of 3; as for 4 skip section 3; 5.1-5; all of 8. Then, dependent upon your desires and or focus as well as your mathematical ability, it should become obvious which of the remaining topics should be read. Finally, the only other source I would recommend for learning Category theory can be found on-line using the keyword 'Awodey'. Anyways, Enjoy and good luck.
36 of 40 people found the following review helpful
Format:Hardcover
This book is extraordinarily well written. It covers the necessary topics in a concise, orderly manner. HOWEVER, it presumes a substantial amount of knowledges concerning various algebraic/abstract structures in the field of mathematics. If you already have had experience with such structures, and are simply looking to understand them from a different perspective - this is the book for you. However, if you have limited knowledge with regards to advanced math (ie - grad level math) then try the book 'Arrows, Structures and Functors: The Categorical Imperative' by Manes and Arbib. This introduces the reader gradually to simple algebraic structures, monoids, groups, metric spaces, topological spaces, and the categories that can be built around them.
Most Recent Customer Reviews
Simply Great
Have you ever tried reading Descartes' "Geometry"? It's not a good place to learn about coordinate geometry. I tried. Read more
Published on January 4, 2008 by J. Wiley
Poorly written standard text.
This book has everything you need, but it is written in an abstruse style in my opinion.
Published on December 7, 2007 by D. Spivak
You may not need this unless you major in category theory.
I entirely agree with the reviewer Lucas Wilman.
As a book by the creator of category theory, it has extensively incorpoated relevant items. Read more
As a book by the creator of category theory, it has extensively incorpoated relevant items. Read more
Published on May 13, 2004
Classic and worth it
It is difficult to make understand what "is" category theory. Is it a foundational discipline? Is it a discipline studying homomorphisms between algebras? Is it nonsense? Read more
Published on August 12, 2003 by Pietro Braione
Kind of Dull
I read this due to its odd title. It is fairly easy to understand. It assumes that you have very little previous knowledge of the subject. For me it just wasn't that useful. Read more
Published on July 4, 2000
Inside This Book
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First Sentence:
Category theory starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams of arrows. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
biproduct diagram, absolute coequalizer, arrows all functions, special adjoint functor theorem, solution set condition, universal arrow, universal cone, limiting cone, dinatural transformation, split coequalizer, triangular identities, comma category, coproduct diagram, general associative law, coherence theorem, canonical arrow, category with finite products, small colimits, adjoint equivalence, strict monoidal category, monoidal categories, function hom, interchange law, composable pair, left adjoint Key Phrases - Capitalized Phrases (CAPs): (learn more)
Mac Lane, Comp Haus New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Category theory starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams of arrows. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
biproduct diagram, absolute coequalizer, arrows all functions, special adjoint functor theorem, solution set condition, universal arrow, universal cone, limiting cone, dinatural transformation, split coequalizer, triangular identities, comma category, coproduct diagram, general associative law, coherence theorem, canonical arrow, category with finite products, small colimits, adjoint equivalence, strict monoidal category, monoidal categories, function hom, interchange law, composable pair, left adjoint Key Phrases - Capitalized Phrases (CAPs): (learn more)
Mac Lane, Comp Haus New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 100
books:
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100
books
cite this book:
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- General Topology (Graduate Texts in Mathematics) by John L. Kelley in Front Matter, and page 190
- Functions of One Complex Variable (Graduate Texts in Mathematics - Vol 11) (v. 1) by John B Conway in Front Matter, and Back Matter
- Elliptic Curves: Diophantine Analysis (Grundlehren der mathematischen Wissenschaften) by S. Lang in Back Matter (1), and Back Matter (2)
- The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics) (v. 106) by Joseph H. Silverman in Back Matter (1), and Back Matter (2)
- Algebraic Topology by William Fulton in Back Matter (1), and Back Matter (2)
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- Mathematical Foundations of Programming Semantics: 9th International Conference, New Orleans, LA, USA, April 7 - 10, 1993. Proceedings (Lecture Notes in Computer Science) by Stephen Brookes on 4 pages
- Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory (Encyclopedia of Mathematics and its Applications) by Maria Cristina Pedicchio on 4 pages
- Current Research in Operational Quantum Logic: Algebras, Categories, Languages (Fundamental Theories of Physics) by Bob Coecke on page 194, page 218, and page 287
- Comprehensive Mathematics for Computer Scientists 2: Calculus and ODEs, Splines, Probability, Fourier and Wavelet Theory, Fractals and Neural Networks, Categories and Lambda Calculus (Universitext) by Guerino Mazzola in Back Matter (1), and Back Matter (2)
- Shape in Picture: Mathematical Description of Shape in Grey-level Images (Nato ASI Series (closed) / Computer and Systems Sciences (closed)) by Ying-Lie O on page 110, and page 145
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